2 problems

Let $N$ be a ground set with $|N|=n$, let $f:2^N\to\mathbb{R}_+$ be a nonnegative monotone submodular function, and let $\boldsymbol{x}\in[0,1]^n$. Let $f^+(\boldsymbol{x})$ be the…

Let $n\geq 2$, let $\boldsymbol{x}\in[0,1]^n$, and let $f:2^{[n]}\rightarrow\mathbb{R}_+$ be a monotone submodular function. Write $f^{+}(\boldsymbol{x})$ for the maximum expected…